Jackson problem 5.4: Magnetic induction in a current-free region

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BREAD
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Homework Statement


Typeset page from a physics problem showing the near-axis series expansions of Bz(rho, z) and B_rho(rho, z) in terms of derivatives of Bz(0, z), followed by the power series form Bz = sum of rho^n b_n(z)/n! and B_rho = sum of rho^n c_n(z)/n!, with B_phi = 0.


Homework Equations

The Attempt at a Solution


I think that is taylor series expansion, How that can be expanded right that? even b0, b1 ... is not a differential term.

Typeset equations from a physics derivation: a coefficient c_{n+1}(z) written as (-1)^{n/2+1} (n+1)! over 2^n (n+2)[(n/2)!]^2 times b_0^{(n+1)}(z), for n odd; then Bz and Bρ series expansions in powers of ρ with k! terms and z-derivatives of Bz(0,z), numbered (4); then equation (5) giving the first terms of Bz and Bρ.

The solution just use b(z) = Bz(0,z)
 
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BREAD said:

Homework Statement


View attachment 203375

Homework Equations

The Attempt at a Solution


I think that is taylor series expansion, How that can be expanded right that? even b0, b1 ... is not a differential term.

View attachment 203376
The solution just use b(z) = Bz(0,z)

Hint: Try taking the divergence and see what you can do with the ##b_{n}## terms algebraically.